Classify the following functions as injection, surjection or bije… - Vidyadip

Question

Classify the following functions as injection, surjection or bijection:
$f : Z \rightarrow Z$, defined by $f(x) = x^2 + x$

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Answer

$f : Z \rightarrow Z$, given by $f(x) = x^2 + x$
Injection test: Let x and y be any two elements in the domain (Z), such that f(x) = f(y).
$f(x) = f(y)$
$x^2 + x = y^2 + y$
Here, we cannot say that $x = y.$
For example, $x = 2$ and $y = -3$
Then, $x^2 + x = 2^2 + 2 = 6$
$y^2 + y = (-3)^2 - 3 = 6$
So, we have two numbers 2 and -3 in the domain Z whose image is same as 6.
Surjection test: Let y be any element in the co-domain (Z), such that f(x) = y for some element x in Z (domain).
$f(x) = y$
$x^2 + x = y$
Here, we cannot say $\text{x}\in\text{Z}$
For example, $y = -4$
$x^2 + x = -4$
$x^2 + x + 4 = 0$
$\text{x}=\frac{-1\pm\sqrt{-15}}{2}=\frac{-1\pm\text{i}\sqrt{15}}{2}$ which is not in Z.
So, f is not a surjection and f is not a bijection.

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